The altitude to the hypotenuse of a 30∘−60∘−90∘ right triangle is divided into two segments of lengths x<y by the median to the shortest side of the triangle. What is the ratio x+yx?
Answer Choices:
A. 73
B. 43
C. 94
D. 115
E. 1543
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Assign a mass of 3 at A, which means we will assign a mass of 1 at C. As AD=DB, this means B has a mass of 3 as well, so E has a mass of 4 and F has a mass of 7.
Thus
EFFB=mass at Bmass at E=34
Thus x=3,y=4, and x+yx=(A)73.
Solution 2:
While there is a simple solution with mass points, we defer to using areas.
Note that
[CEF][CFB]=21⋅EB⋅distance from C to EF21⋅FB⋅distance from C to FB=EFFB
as FB and EF are the same line. Note that
[CAF][CEF]=21⋅FE⋅CA21⋅FE⋅CE=CACE
as FE is perpendicular to AC. Note that
[CAD]=21⋅AD⋅CB=21⋅BD⋅CB=[CBD]
[FAD]=21⋅AD⋅distance from F to AD=21⋅BD⋅distance from F to BD=[FBD]
so [AFC]=[ACD]−[AFD]=[BCD]−[BFC]=[CFB]. Thus, combining these we get
